1remaining Figure I S L A. Becauſe now that N O is Seſquialter
of R O, but leſs than Seſquialter ejus quæ uſque ad Axem (or of its
Semi-parameter;) ^{*} R O ſhall be leſſe than quæ uſque ad Axem (or
than the Semi-parameter;) ^{*} whereupon the Angle R P ω ſhall be
acute. For ſince the Line quæ uſque ad Axem (or Semi-parameter)
is greater than R O, that Line which is drawn from the Point R,
and perpendicular to K ω, namely RT, meeteth with the line F P
without the Section, and for that cauſe muſt of neceſſity fall be
tween the Points P and ω; Therefore if Lines be drawn through
B and G, parallel unto R T, they ſhall contain Right Angles with
the Surface of the Liquid: ^{*} and the part that is within the Li
quid ſhall move upwards according to the Perpendicular that is
drawn thorow B, parallel to R T, and the part that is above the Li
quid ſhall move downwards according to that which is drawn tho
row G; and the Solid A P O L ſhall not abide in this Poſition; for
that the parts towards A will move upwards, and thoſe towards
B downwards; Wherefore N O ſhall be conſtituted according to
the Perpendicular.]
of R O, but leſs than Seſquialter ejus quæ uſque ad Axem (or of its
Semi-parameter;) ^{*} R O ſhall be leſſe than quæ uſque ad Axem (or
than the Semi-parameter;) ^{*} whereupon the Angle R P ω ſhall be
acute. For ſince the Line quæ uſque ad Axem (or Semi-parameter)
is greater than R O, that Line which is drawn from the Point R,
and perpendicular to K ω, namely RT, meeteth with the line F P
without the Section, and for that cauſe muſt of neceſſity fall be
tween the Points P and ω; Therefore if Lines be drawn through
B and G, parallel unto R T, they ſhall contain Right Angles with
the Surface of the Liquid: ^{*} and the part that is within the Li
quid ſhall move upwards according to the Perpendicular that is
drawn thorow B, parallel to R T, and the part that is above the Li
quid ſhall move downwards according to that which is drawn tho
row G; and the Solid A P O L ſhall not abide in this Poſition; for
that the parts towards A will move upwards, and thoſe towards
B downwards; Wherefore N O ſhall be conſtituted according to
the Perpendicular.]
COMMANDINE.
The Demonſtration of this propoſition hath been much deſired; which we have (in like man
ner as the 8 Prop. of the firſt Book) reſtored according to Archimedes his own Schemes, and
illustrated it with Commentaries.
ner as the 8 Prop. of the firſt Book) reſtored according to Archimedes his own Schemes, and
illustrated it with Commentaries.
The Right Portion of a Rightangled Conoid, when it ſhall
have its Axis leſſe than Seſquialter ejus quæ uſque ad Axem (or of
its Semi-parameter] In the Tranſlation of Nicolo Tartaglia it is falſlyread great
er then Seſquialter, and ſo its rendered in the following Propoſition; but it is the Right
Portion of a Concid cut by a Plane at Right Angles, or erect, unto the Axis: and we ſay
that Conoids are then conſtituted erect when the cutting Plane, that is to ſay, the Plane of the
Baſe, ſhall be parallel to the Surface of the Liquid.
have its Axis leſſe than Seſquialter ejus quæ uſque ad Axem (or of
its Semi-parameter] In the Tranſlation of Nicolo Tartaglia it is falſlyread great
er then Seſquialter, and ſo its rendered in the following Propoſition; but it is the Right
Portion of a Concid cut by a Plane at Right Angles, or erect, unto the Axis: and we ſay
that Conoids are then conſtituted erect when the cutting Plane, that is to ſay, the Plane of the
Baſe, ſhall be parallel to the Surface of the Liquid.
Which ſhall be the Diameter of the Section I P O S, and the
Axis of the Portion demerged in the Liquid.] By the 46 of the firſt of
the Conicks of Apollonious, or by the Corol
lary of the 51 of the ſame.
Axis of the Portion demerged in the Liquid.] By the 46 of the firſt of
the Conicks of Apollonious, or by the Corol
lary of the 51 of the ſame.
21[Figure 21]And of the Solid Magnitude A P
O L, let the Centre of Gravity be R;
and of I P O S let the Centre be B.]
For the Centre of Gravity of the Portion of a Right
angled Conoid is in its Axis, which it ſo divideth
as that the part thereof terminating in the vertex,
be double to the other part terminating in the Baſe; as
in our Book De Centro Gravitatis Solidorum Propo. 29. we have demonſtrated. And
ſince the Centre of Gravity of the Portion A P O L is R, O R ſhall be double to RN and there
fore N O ſhall be Seſquialter of O R. And for the ſame reaſon, B the Centre of Gravity of the Por
tion I P O S is in the Axis P F, ſo dividing it as that P B is double to B F;
O L, let the Centre of Gravity be R;
and of I P O S let the Centre be B.]
For the Centre of Gravity of the Portion of a Right
angled Conoid is in its Axis, which it ſo divideth
as that the part thereof terminating in the vertex,
be double to the other part terminating in the Baſe; as
in our Book De Centro Gravitatis Solidorum Propo. 29. we have demonſtrated. And
ſince the Centre of Gravity of the Portion A P O L is R, O R ſhall be double to RN and there
fore N O ſhall be Seſquialter of O R. And for the ſame reaſon, B the Centre of Gravity of the Por
tion I P O S is in the Axis P F, ſo dividing it as that P B is double to B F;